Original Citation:
Symmetrizing quantum dynamics beyond gossip-type algorithms
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10.1016/j.automatica.2016.06.019
Università degli Studi di Padova
Symmetrizing Quantum Dynamics
Beyond Gossip-type Algorithms
Francesco Ticozzi
a,ba
Dipartimento di Ingegneria dell’Informazione, Universit`a di Padova, via Gradenigo 6/B, 35131 Padova, Italy
b
Deptartment of Physics and Astronomy, Dartmouth College, 6127 Wilder, Hanover, NH (USA)
Abstract
Recently, consensus-type problems have been formulated in the quantum domain. Obtaining average quantum consensus consists in the dynamical symmetrization of a multipartite quantum system while preserving the expectation of a given global observable. In this paper, two improved ways of obtaining consensus via dissipative engineering are introduced, which employ on quasi local preparation of mixtures of symmetric pure states, and show better performance in terms of purity dynamics with respect to existing algorithms. In addition, the first method can be used in combination with simple control resources in order to engineer pure Dicke states, while the second method guarantees a stronger type of consensus, namely single-measurement consensus. This implies that outcomes of local measurements on different subsystems are perfectly correlated when consensus is achieved. Both dynamics can be randomized and are suitable for feedback implementation.
Key words: Quantum control, Consensus, Distributed control, Symmetrization, Robust control
1 Introduction
In quantum as in classical systems, symmetry is a fundamental concept, and a key tool in investigating dynamical systems. In particular, suitable dynamical symmetries, or equivalently the existence of preserved quantities, prevent controllability for quantum dynam-ics [2,6,8], while they allow for the existence of protected sets of states [16,15,28,27]. Symmetric quantum states and subspaces have a key role in the description of quantum systems obeying Bose-Einstein statistics [14], they are related to thermalization [10], and have a key role in quantum information [21]. Prototypical states used to illustrate and exploit intrisecally quantum cor-relations between information units, or entanglement between qubits in the quantum information jargon, are the maximally entangled states named after Greenberg-Horn-Zeilinger (GHZ) [12] and the W states [9]: both are invariant with respect to permutation of their sub-systems.
? This paper was not presented at any IFAC meeting. Cor-responding author F. Ticozzi. Tel. +39-049-8277603. Fax +39-049-8277614.
Email address: [email protected] (Francesco Ticozzi).
In [20] it is shown that a class of classical dynamics obtaining asymptotic consensus can be seen as sym-metrizing dynamics, leading to permutation-invariant states. The same underlying idea led to dynamics for symmetrizations in the quantum realm [19]. There, it is shown how quantum consensus algorithms can be used in combination with simple local controls and measure-ments in order to prepare pure states and estimate the size of a network. This type of symmetrizing dynam-ics and their convergence properties, as well as their continuous-time counterpart, have been further studied in [24,22,23]. All the symmetrizing dynamics that have been proposed so far, however, are based on combina-tions of permutation operators, and hence they share two common properties: (1) they are unital, i.e. the max-imally mixed state is preserved; (2) they attain sym-metric state consensus, which is effectively consensus on the statistical properties of the variables of interest, but there is no algorithm that can attain actual consensus on the output of each local measurement. Due to the contraction properties of the considered maps, unital dy-namics entails that the purity of the quantum state can not be augmented by the consensus-reaching dynam-ics. With a degradation of purity, the fragile quantum correlations encoded in the state are typically lost, and stronger notions of consensus are out of reach. If the consensus-achieving dynamics improve purity, they can
be instrumental to distributed and robust preparation of interesting states, like Dicke (including W) or GHZ states [25,26], as we shall demosntrate in the following.
In the effort of overcoming these issues, two types of symmetrizing dynamics are proposed, which allow for asymptotic symmetrization of the state of a multipartite quantum system with respect to the permutation group acting on the subsystems. Both dynamics are composed of quasi-local maps and are robust with respect to ran-domization, and are hence suitable for distributed and unsupervised implementations – as are their classical and classically-inspired counterparts.
The first one attains an asymptotically symmetric state while preserving the expectation value of a global ob-servable, reaching average symmetric state consensus in the language of [19]. It does so by selecting and prepar-ing a pure symmetric state (a Dicke state [7,11]) in each eigen-subspace of the observable of interest. It is also shown how these dynamics are instrumental to prepa-ration of globally entangled pure states. The proposed protocol uses only single system operations and pairwise interactions.
The second dynamics is the first proposal of a quasi-local protocol that obtains a stronger type of quantum con-sensus, called single-measurement consensus [19]. This type of consensus, which implies the symmetry of the state but for which the latter is not sufficient, is the closest in spirit to classical consensus: after single mea-surement consensus is reached, the meamea-surement of a lo-cal observable quantity on any subsystem will force the whole network of systems to “agree” on the result, i.e. yielding perfectly correlating results. Notice that, in con-trast with classical consensus, the consensus value may not be determined before it is actually measured.
Both methods rely on essentially quantum features of the system and, while obtaining a symmetric pure state that has a specified average of an observable is not viable in general, they allow for a final purity that is typically better than the one offered by gossip-type algorithms. Both are suitable for implementation via discrete-time feedback [4], and can be combined with local initializa-tion procedures in order to actually prepare perfectly pure and entangled symmetric states. This is explicitly shown for the first proposed algorithm (see Corollary 2). These dynamics can be seen as the discrete-time equivalent of conditional preparation of entangled states, in the spirit of [26].
The paper structure is as follows: in Section 2 a brief review of the relevant quantum consensus definitions and the ideas underlying existing consensus-achieving dynamics is provided; Section 3 begins by explaining why trying to obtain quantum consensus with just pure states is impossible, and continues by providing the form
and the convergence proofs for two novel symmetriz-ing dynamics. While these can be straightforwardly ex-tended to networks of d-dimensional systems, as those considered in the introductory Section, the presentation here focuses on qubit networks and pairwise interactions. Qubit systems are easier to visualize and yet relevant for applications, some of which have been discussed in [19]. Pairwise interactions are the minimal that can be allowed for interacting dynamics, and if a more forgiv-ing locality constraint is in place, a set of effective pair-wise interactions will also be allowed under this locality notion. Section 4 illustrates the behavior of the two dy-namics, and compares them with the existing consensus algorithm.
2 Background
2.1 Quantum Consensus States
Consider a multipartite system composed of m isomor-phic subsystems, labeled with indices i = 1, . . . , m, with associated Hilbert space Hm:= H
1⊗ · · · ⊗ Hm' H⊗m, with dim(Hi) = dim(H) = n and 2 6 n < ∞. This mul-tipartite system will act as our quantum network. We shall use Dirac’s notation: |ψi denote vectors of H, hψ| denote their dual linear functional. B(H) denotes the set of linear operators on H, which in our finite-dimensional setting are in a one-to-one relationship with complex ma-trices. States are associated to density operators, namely linear, trace-one, positive-semidefinite operators on H, with their set denoted by D(H) ⊂ B(H). Observable quantities can be associated to Hermitian operators on H, denoted by H(H). The support of an Hermitean op-erator is the orthogonal complement to its kernel. Given a state ρ and an observable X, the expectation of X ac-cording to ρ is computed as Tr(ρX).
For any operator X ∈ B(H), denote by X⊗mthe tensor product X⊗X⊗...⊗X with m factors. Given an operator σ ∈ B(H), denote by σ(i)the local operator:
σ(i):= I⊗(i−1)⊗ σ ⊗ I⊗(m−i).
Permutations of quantum subsystems are expressed by a unitary operator Uπ∈ U(H), which is uniquely defined by
Uπ(X1⊗ . . . ⊗ Xm)Uπ† = Xπ(1)⊗ . . . ⊗ Xπ(m) for any operators X1, . . . Xmin B(H), where π is a per-mutation of the first m integers. P is the set of such π. A state or observable is said to be permutation invariant if it commutes with all the subsystem permutations.
In [19] a number of potential extensions of the idea of classical consensus to a quantum network were proposed, and their merit discussed in depth. The ones relevant to this work are:
Definition 1 (SSC) A state ρ ∈ D(Hm) is in Symmet-ric State Consensus (SSC) if, for each unitary permuta-tion Uπ,
Uπρ Uπ†= ρ .
Definition 2 (σSMC) Given σ ∈ B(H) with spectral decomposition σ = Pd
j=1sjΠj ∈ H(H), a state ρ ∈ D(Hm) is in Single σ-Measurement Consensus (σSMC) if:
Tr(Π(k)j Π(`)j ρ) = Tr(Π(`)j ρ), (1) for all k, ` ∈ {1, . . . , m}, and for each j.
The definition of σSMC is the unique, among those pro-posed, that requires that the outcomes of σ measure-ments on different subsystems be exactly the same for each trial. Consider the set of projections {Πj}dj=1as in Definition 2, and let us define
ΠSMC= d X j=1
Πj⊗m.
It has been shown that a state is in σSMC if and only if it holds
Tr(ΠSMCρ) = 1, (2) or equivalently
ΠSMCρΠSMC= ΠSMCρ = ρ. (3) Furthermore σSMC for σ with non-degenerate spectrum implies SSC, while the converse implications do not hold. Lastly, it is impossible for a state to be σSMC with re-spect to all σ ∈ H(H): this means that σSMC cannot be strengthened to a single-measurement equivalent of SSC. The proofs of these statements are given in [19].
It is worth remarking how all these definitions could be given for classical systems, in the context of consensus for random variables or for probability distributions of the state values. In some sense the definition of σSMC is the closest to the classical case, as it requires perfect agree-ment on the outcome of a set of random variables for each measurement. However, in contrast with the clas-sical “perfectly correlated” random variables, the quan-tum version allows for the output to be determined only at the moment of the measurement. Both SSC and SMC (with non-degenerate observable σ) imply that the final state is permutation invariant.
2.2 Quantum Average Consensus
So far, only “static” properties of consensus states have been discussed. However, the core of the problem of in-terest for this paper is the design of discrete-time dynam-ical systems (or, from an information processing perspec-tive, algorithms) that drive the system to consensus. In
addition to this, as in the classical consensus problems, the final state will be required to preserve or express some property dependent on the initial state.
Unitary dynamics are not enough when ones is inter-ested in studying or engineering convergence features for a quantum system. A more general framework that in-cludes (Markovian) open-system evolutions is offered by quantum channels [17,21], that is, linear, completely pos-itive (CP) and trace preserving (TP) maps from density operators to density operators E : D(Hm) → D(Hm). It can be shown that such maps admit an operator sum rep-resentation (OSR), also known as Kraus decomposition:
E(ρ) = K X k=1 AkρA†k with K X k=1 A†kAk= I (4)
where K 6 (dim(H))2. The representation is not unique, however the relation between all the possible different representations is well known (see [21, Theorem 8.2]). A CPTP map is said unital if E (I) = I. Given a CPTP map E , its dual map with respect to the Hilbert-Schmidt inner product E†is defined by:
Tr[A E (ρ)] = Tr[E†(A) ρ] . (5) This dual map is still linear and completely positive, while the fact that E is trace preserving implies that E† is unital.
Locality notions for the quantum network can be intro-duced as it follows [25]. Consider the multipartite sys-tem introduced in Section 2.1. An operator V is a neigh-bourhood operator with respect to a set of neighborhoods {Nj , j = 1, 2, ..., M }, if there exists j ∈ {1, 2, ..., M } such that:
V = VNj ⊗ INj (6)
where VNj accounts for the nontrivial action on HNj and
IN
j =
N k /∈NjIk.
Everything is in place to specify the consensus prob-lems of interest. In addition to asymptotically achieving a state with the prescribed “informational” symmetry, this state must maintain locally some information on the global initial state. Let d(ρa, C) = infρ∈Ckρa−ρk, where C ⊂ D(H) and k · k is any p-norm on B(H). Given a se-quence of QL channels {Et(·)}∞t=0, define ˆEt(ρ0) = ρt= Et◦ Et−1◦ · · · ◦ E1(ρ0), and CSSC to be the set of states in SSC consensus, and similarly for σSMC. Let S be an observable in B(H⊗m).
Definition 3 (Asymptotic S-Average Consensus) A sequence of channels {Et(·)}∞t=0, is said to asymptoti-cally achieve SSC if
lim
for all initial states ρ0, and there exists a σ ∈ H(H) such that: lim t→∞Tr(σ (`)ρ(t)) = lim t→∞Tr(Sρ(t)) = Tr(Sρ0) (8) for all ρ0and for all ` ∈ {1, . . . , m}.
The same definition holds for σSMC by substituting the corresponding state sets in (7).
2.3 Gossip-type Dynamics
Let us recall the structure and core results for the gossip-type algorithm introduced in [19]. In a controlled quan-tum network, one can typically engineer unitary trans-formations that implement the “identity” evolution and the swapping of two neighboring subsystem states; let us denote the latter operator by U(j,k)for swapping sub-systems j and k. To develop our analysis, it will be con-venient to introduce the graph G associated to the mul-tipartite system: its nodes 1, . . . , m correspond to the “physical” subsystems, the edge (j, k) is included if the subsystems j and k (have a non-zero probability to) in-teract.
Assume edge (j, k) is selected at a certain step t: then implement on subsystems j, k the quantum gossip inter-action: ρ(t + 1) = Ej,k(ρ(t)) = (1 − α) ρ(t) + α U(j,k)ρ(t)U † (j,k), (9) with α ∈ (0, 1).
It has been proven that, if the graph associated to pos-sible interactions is connected, then the quantum gossip algorithm (9) ensures global convergence towards:
ρ∗= 1 m! X π∈P Uπρ0Uπ† ∈ CSSC. (10)
Convergence is deterministic, when the edges on which a gossip interaction occurs at a given time are selected by periodically cycling, in any predefined way, through the set of available ones; or with probability one1, and thus in expectation, when the edges on which a gossip interac-tion occurs at a given time are selected randomly from a fixed probability distribution {qj,k > 0|P(j,k)∈Eqj,k = 1}; Furthermore, S-average SSC is attained if and only
1 This means that, for any δ, ε > 0, there exists a time T > 0
such that
P[ Tr((ρ(T ) − ρ∗)2) > ε ] < δ .
if S ∈ H(H⊗m) can be written, for some σ ∈ H(H), in the form: S = 1 m m X i=1 σ(i). (11) This shows that the mean value of a (global) observable can be asymptotically retrieved from the state of any single subsystem by employing a quantum gossip-type algorithms.
3 New Results
3.1 Preliminary observations
The dynamics associated to the maps (9) are always unital, that is the identity is always a fixed point for the dynamics. Given the (trace-norm) contraction character of CPTP evolutions [1], even if the state asymptotically converges to a different state, the evolution is bound to get closer to the identity, i.e. the completely mixed state.
This type of symmetrization, albeit interesting and sat-isfying the requisites for asymptotic average SSC, is pu-rity decreasing and in some sense makes the system state “more classical”. The natural question is then: can quan-tum consensus be achieved with a pure state?
This, in general, is not possible. The desired dynamics should select a pure state for any output, while preserv-ing the expectation value of the desired global observ-able: it would entail a linear map E such that:
Tr(SE (ρ)) = Tr(Sρ), (12)
and TrE (ρ)2) = 1. Write S in spectral decomposition as S =P
kλkΠk, with Πk the orthogonal projectors onto the (degenerate) eigenspaces Vkof S. For any pure state |ψki ∈ Vk, the output E (|ψkihψk|) = |φkihφk| should remain pure. However, any linear map E would satisfy
E(λk|ψkihψk| + (1 − λ)|ψjihψj|)
= λkE(|ψkihψk|) + (1 − λ)E(|ψjihψj|) = λk|φkihφk| + (1 − λj)|φjihφj|, where 0 < λ < 1. The only case in which the output can be pure for any λ is when E (ψkihψk|) = E(|ψjihψj|) : this would satisfy the expectation preservation if and only if |ψ1i and |ψ2i had the same expectation, which is clearly not possible for j 6= k. Inspired by this observation, we will construct dynamics that select a pure representative in each of the eigenspaces of the global observable S, yet losing some of the correlations between subspaces.
In the rest of the paper, in order to limit the no-tational complexity and provide some matrix rep-resentations that are easier to visualize, the focus is on: (1) networks of two-level systems or qubits,
Hi = span{|0i, |1i}. We use the common shorthand notation for the standard basis of a composite Hilbert space Hm: |01 . . .i = |0i ⊗ |1i ⊗ . . .; (2) pair-wise inter-actions, i.e. the neighborhoods are a subset of the set of pairs {{j, k}, 1 ≥ j, k ≤ m, j 6= k}. Both assumptions can be relaxed, with some straightforward adaptation in the algorithms.
3.2 Symmetrizing dynamics for symmetric state con-sensus
The preservation of the asymptotic expectations is clearly guaranteed if each eigenspace of S is an invari-ant subspace for all maps that can enter the evolution, at any time. More precisely, if S is written in spectral decomposition as S = P
kλkΠk, with Πk the orthogo-nal projectors onto the (degenerate) eigenspaces of S, call them Hk, invariance of all the Hk is equivalent to require E†(Πk) = Πk [5], so that:
Tr(Et(S)ρ) = Tr( X
k
λkEt(Πk)ρ) = Tr(Sρ).
On the other hand, as long as convergence to a con-sensus state is guaranteed, dynamics inside the sub-space is arbitrary. Let us build one that, while satisfying eigenspace invariance, prepares a pure representative in each S eigenspace. For networks of two-level systems, up to a local change of basis and a rescaling, any S as in (11) can be rewritten as S = mI + m X i=1 σz(i). (13)
This allows to identify the eigenspaces of S with respect to this basis as the “excitation” subspaces:
Hk = span{|i1, i2, . . . , imi, ik∈ {0, 1}, m X `=1 i`= k} (14) generated by linear combination of vectors belonging to the standard basis associated to exactly k 1’s. As typical in the physics literature, call the number of 1’s in such vectors, and thus of the generated subspaces, the excita-tion number. Each of these subspaces contains only one pure symmetric state, which is referred in the physics literature as a Dicke state [7,11]. Each state is labeled by the number of systems N and the number of excitations k, and is associated to a vector:
|(m, k)i = q1 m k (| 1 . . . 1 | {z } k 0 . . . 0 | {z } m−k i + . . . + | 0 . . . 0 | {z } m−k 1 . . . 1 | {z } k i), (15) and each term of the sum corresponds to a unique ar-rangement of the k ones and N − k zeros. To our aim,
the key property of Dicke states, beside being pure and symmetric, is that they admit a very simple Schmidt de-composition. Consider a partition of the network into two groups of mA and mB subsystems, respectively. It is then easy to show that the Schmidt decomposition of a Dicke state |(m, k)i is
|(m, k)i = q1 m k X kA+kB=k µkA,kB|(mA, kA)i⊗|(mB, kB)i, (16) where the weight on each term is µkA,kB =
q m
A
kA
mB
kB.
In order to explicitly construct the dynamics that pre-pare these states in each eigenspace, choose a neighbour-hood Nj, containing 2 qubits, and consider the neigh-bourhood operator:
SNj = 2I +
X i∈Nj
σ(i)z . (17)
As we did for the global S in (14), we can construct the eigenspaces SNj as the subspaces spanned by pure states
with a given number of excitation:
HNj k = span{|i1, i2, . . . , imji, i`∈ {0, 1}, mj X `=1 i`= k, } (18) which, in our two-qubit neighborhood, are just three:
HNj 0 = span{|00i}, HNj 1 = span{(|01i + |10i)/ √ 2, (|01i − |10i)/√2}, HNj 2 = span{|1, 1i}. (19) Joining these sets of vectors (in order), a basis for HNj
0 ⊕ HNj
1 ⊕ H Nj
2 is obtained. With respect to this basis, con-struct a neighbourhood CPTP map ˜ENj with operators:
˜ Mj,1= 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 (20) ˜ Mj,2= 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 . (21)
From these neighborhood operators, it is easy to con-struct a map on the whole network, which is going to
be the dynamics associated to the selection of neighbor-hood Nj:
ENj(ρ) = ˜ENj ⊗ IN¯j(ρ) (22)
= ˜Mj,1⊗ I ρ ˜Mj,1† ⊗ I + ˜Mj,2⊗ I ρ ˜Mj,2† ⊗ I. Here IN¯j is the identity map on B(HN¯j), which is the set of operators that are forced to evolve trivially for neighborhood maps on Nj. Now notice that each map is CPTP, and:
(1) The first operator is nilpotent and acts like a gen-eralized “raising” operator, which ensures that in-variant neighbourhood states for the map ˜ENj must
have support on
H0j = span{|(2, `)i, ` = 0, 1, 2}, which is the kernel of ˜M1,j.
(2) The second operator ˜Mj,2 is just the orthogonal projector onto H0j. The latter is thus an invariant subspace for ˜ENj, and any state with support on it
is a fixed point.
(3) Given the ladder structure of ˜M1,j, any state for the neighborhood that does not have support on it is attracted to H0
j.
In terms of typical available gates/control resources for pairs of qubits, such a map can be implemented in a “digital quantum simulator” [3], with a two-body uni-tary (that e.g. maps the ordered standard basis into {(|0i+|1i)/√2, (|0i−|1i)/√2, |00i, |11i}), a single qubit reset and another two body unitary reverting the effect of the first. These maps can also be implemented via dis-crete quantum feedback [4], via a direct projective mea-surement with two outcomes, associated to projectors {Π1= I − ˜Mj,2, Π2= ˜Mj,2}, followed by a unitary oper-ation when the outcome associated to the first projector is observed: ˜ Mj,1= U1Π1, U1= 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 ,
where the matrix representation is given with respect to the basis constructed above.
All is in place to prove the following:
Theorem 1 If the neighborhoods cover the whole net-works and the associated graph2 is connected, global con-vergence towards SSC is ensured:
2
The graph that has subsystems as nodes, and edges
con-- Deterministically, when all the neighborhoods in any order such that all neighborhoods are selected an infinite number of times;
- In probability and in expectation, when all the neigh-borhoods are selected randomly from a fixed probability distribution {qj = P[Nj] > 0|Pjqj = 1}. In any of the above cases, the system converges to a state ρ∗ that has support on
H0= span{|(m, `)i, ` = 0, . . . , m}. (23) Furthermore, if the initial state has support on the global eigenspaces of S with eigenvalue λ, the final state is the pure state ρ∗= |(m, λ)ih(m, λ)|.
Proof. Let H0 be the span of the Dicke states on the whole network. From decomposition (16), it follows that for each j, it holds that:
H0⊆ H0 j ⊗ H 0 ¯ Nj. (24) Hence H0 ⊆ T jH 0 j ⊗ H 0 ¯
Nj and it is invariant for each
map as well. It is easy to see that H0is actually H0 = T
jH 0 j ⊗ H0N¯
j. In fact, if a vector is not in H
0, then it is also not in H0j for some j. If a state ρ is invariant for all maps ENj, then it has support on the symmetric
sub-space. Since they do not change the excitation number, each ENj leaves the eigenspaces of the corresponding SNj
invariant, as well as those of the global S.
Consider now a finite sequence of T neighborhood maps {ENj(t)}t=1,2,...,T in which enough overlapping
neighbor-hood are chosen so that they cover the whole network of m qubits, and the define the composite map:
Emo = ENj(T ) ◦ . . . ◦ ENj(1).
Since each ENj(t) leaves the eigenspaces of S invariant,
the same is true for Eo m.
Consider the invariant subspaces Hk of S, and define Vkm(ρ) = 1−h(m, k)|ρ|(m, k)i, and the (m-system) Dicke density operators as
ρm,k= |(m, k)ih(m, k)|. (25) Clearly Vkm(ρm,k) = 0, and Vkm(ρ) > 0 if ρ 6= ρm,k, and it will be our Lyapunov-type functions: if it is true that for all k and ρ 6= ρm,k with support in Hk there is a sequence of maps such that:
∆Vkm(ρ) := Vkm(Emo(ρ)) − Vkm(ρ) < 0, (26)
necting pairs of subsystems that belong to same neighbor-hood.
then the only state with support on Hk such that ∆Vm
k (ρ) = 0 would be ρm,k. Hence, considering V (ρ) =P
kV m
k (ρ), the main Theorem statement would follow by the discrete-time version of the invariance principle (see Appendix A), since we would prove con-vergence to H0, and it would be possible to further conclude that ρm,kis asymptotically stable in D(Hk). The key is thus to prove that there exists a sequence such that (26) holds, by induction on the number of qubits m. The inductive hypothesis is: For m qubits and all 0 ≤ k ≤ m, there exists Emo such that ∆Vkm(ρ) < 0 for any ρ ∈ D(Hm,k), ρ 6= ρm,k.
For m = 2, the network consists in a single pair, so there is a single neighborhood N = {1, 2}, and Eo
2 = EN. By construction, EN satisfies the inductive hypothesis. In fact,
HN0 = span{|0, 0i}, HN2 = span{|1, 1i}
only support the invariant states ρ2,0, ρ2,2, respec-tively, while on HNj
1 it prepares the state ρ2,1 = (|0, 1i + |1, 0i)(h0, 1| + h1, 0|)/2 with a single application, so ∆V1m(ρ) = −V1m(ρ) < 0.
Next, the inductive step is proved. Choose a partition of the m+1 qubits in 2 qubits and the remaining m−1 ones: without loss of generality (up to a relabelling), choose N1= {1, 2} andN1. Consider a Em+1o of the form EN1◦
Eo
m. Since ρm+1,kis invariant for all neighborhood maps, both Vkm+1(Eo m(ρ))−V m+1 k (ρ) ≤ 0 and V m+1 k (EN1(ρ))−
Vkm+1(ρ) ≤ 0 for any initial ρ [4]. It is next shown that at least one of them is strictly negative if ρ 6= ρm+1,k. If it holds already Vkm+1(Emo(ρ)) − Vkm+1(ρ) < 0, then the induction step is already proved. Recall that ρm+1,k is the projector onto
span{|(m + 1, k)i} = span|0i ⊗ |(m, k)i +|1i ⊗ |(m, k − 1)i = span|00i ⊗ |(m − 1, k)i
+(|01i + |10i) ⊗ |(m − 1, k − 1)i +|11i) ⊗ |(m − 1, k − 2)i ,
and
H(m+1,k)= span|0i} ⊗ H(m,k)⊕ span{|1i ⊗ H(m,k−1). By the inductive hypothesis and conservation of the excitation number k, it follows that Emo(ρ) guarantees ∆Vkm+1< 0 for any state on H(m+1,k), unless:
supp(ρ) ⊆ (span{|0i ⊗ |(m, k)i}) ⊕ (span{|1i ⊗ |(m, k − 1)i}) .
Since ρm,kis invariant for all neighborhood maps, then considering Em+1o (ρ) = ENk◦ E
o
m(ρ) still makes Vkm de-crease.
Consider now the case in which Emo leaves Vkm unvar-ied and let us consider an additional map EN1, with the
neighborhood overlapping, but not being included, with the subsystems non-trivially affected by Eo
m(ρ). In terms of the N1,N1partition, this means:
supp(ρ) ⊆ Hinv
= (span{|00i ⊗ |(m − 2, k)i + |01i ⊗ |(m − 2, k − 1)i}) ⊕ (span{|10i ⊗ |(m − 2, k − 1)i + |11i ⊗ |(m − 2, k − 2)i})
The action of EN1 maps the subspace orthogonal to the
target in Hinv, namely the span of: |φ⊥i = |00i ⊗ |(m − 1, k)i
+(|01i − |10i) ⊗ |(m − 1, k − 1)i −|11i ⊗ |(m − 1, k − 2)i
to
span|00i ⊗ |(m − 1, k)i
+(|01i + |10i) ⊗ |(m − 1, k − 1)i −|11i ⊗ |(m − 1, k − 2)i.
This implies that either ρ = ρm+1,kor:
Vkm+1(EN1(ρ)) − V m+1 k (ρ) = − 1 2hφ ⊥|ρ|φ⊥i < 0.
Notice that is shows it is sufficient to choose the sequence of neighbourhoods, and thus the sequence of maps, so that they have always overlap with the already selected ones and eventually cover the whole network. This is also shown by induction on m: for m = 2 it is trivially true with Eo
2 as no other choices of neighborhoods is possi-ble. If it is true for m, and the graph associated to the neighborhoods is connected, any sequence of neighbor-hood that satisfies the above properties on some con-nected sub-graph of m qubits and adding an additional two-bit neighborhood that covers the m + 1-th qubit can be used. The proof above applies to this case, and the Lyapunov function strictly decreases for sequences con-structed in this way.
Convergence in probability is then proved by a direct application of Lemma 1 in the Appendix, using the se-quence associated to Eo
m, and convergence in expectation trivially follows from convergence in probability. 2 From Theorem 3, it is easy to prove the following result.
Corollary 2 If the following control capabilities are available:
(i) Projective measurements of σz and unitary control σxfor the single qubits;
(ii) Pairwise interactions as in (22) on a connected graph;
Then any Dicke state is asymptotically preparable. If, in addition, projective measurements of S as in (13) are available, any Dicke state can be made asymptotically stable.
Proof. Consider a target Dicke state, and assume it is in the `-eigenspace of S. The strategy to prepare it is the following:
(1) First, we want to prepare the correct eigenspace of S. To do so, if this additional resource is available, first measure S: if the outcome is `, move to the next step. Otherwise, if either we cannot measure S or the eigenspace is not the correct one, start measuring the single qubits. Let `0 be the number of 1 outcomes. If ` − `0= q > 0 apply σ
xto q qubits that gave −1 as outcome, if q < 0 apply σz to q qubits that gave 1 as outcome. This will prepare a pure factorized, yet not symmetric, state in the correct eigenspace.
(2) Next, run the SSC algorithm of Theorem 3 using any cyclic repetition of overlapping pairwise inter-actions that cover the whole network.
By the previous result, we will have convergence to |(m, `)ih(m, `)|. Notice that if S can be measured and the initial state is already the correct Dicke state, it will remain invariant for the dynamics, as single qubit measurements will not be applied. Hence, the above protocol effectively asymptotically stabilizes the target. On the other hand, if S cannot be measured, the single qubit measurements will initially disrupt the desired state, which will be re-prepared by the second step, as in quasi-local conditional stabilization procedures of
[25]. 2
This can be thought as a discrete-time, generalized ver-sion of the strategies proposed to conditionally stabilize W states with continuous semigroup dynamics in [25]. Dicke states, with the exception of the ones correspond-ing to extremal eigenspaces, are entangled and of inter-est for quantum information applications [21].
3.3 Symmetrizing Dynamics for Single-Measurement Consensus
The following dynamics addresses average σSMC. Given Theorem 1, it is necessary to design QL dynamics that drive the state on the support of the projector ΠSMC, while maintaining the expected value of S. In the qubit network case, after putting S in the standard form (13), the support of ΠSMCcorresponds to:
HSMC= span{|00 . . . 0i, |11 . . .i}.
In order to construct the stabilizing neighborhood maps, consider the standard basis {|kji}
dNj −1
k=0 for HNj, where
|0i = |00 . . . 0i, |1i = |00 . . . 01i, and so on with binary ordering until |dNj−1i = |11 . . . 1i.
Define ΠNj,k = |kjihkj| and ΠNj,sym = ΠNj,1 +
ΠNj,dNj −1. This is a projector on the neighborhood
re-duction of HSMC, namely HNj,SMC. Now construct a
neighbourhood CPTP map: ˜ ENj(ρ) = ΠNj,symρΠNj,sym+ dNj −2 X k=1 ˜ RNj,k(ΠkρΠk), (27) where: ˜ RNj,k(τ ) = dNj −2 X k=1 pk,0Uk,0τ Uk,0† + pk,1Uk,1τ Uk,1† , and:
(1) Uk,0 satisfies Uk,0|ki = |0i and Uk,1 satisfies Uk,1|ki = |dNj−2i;
(2) pk,0 is equal to the number of zeros in the binary representation of k divided by the number of qubits in the neighborhood, and pk,1= 1 − pk,0.
Intuitively, E˜Nj leaves any state with support on
HNj,SMC invariant, while on the orthogonal it performs
a measurement projecting the state on the standard ba-sis for the neighborhood, followed by a “randomizing” map ˜RNj,k. This map transfers each basis vector to a
mixture of |00 . . . 0i, |11 . . . 1i, with weights pk,0, pk,1 chosen so that they preserve the expectation of S.
The maps on the whole network are obtained by the neighborhood maps as:
ENj(ρ) = ˜ENj ⊗ IN¯j(ρ) (28)
By construction, each of these maps is CPTP, and leave any state on HSMCinvariant. The following convergence result holds:
Theorem 3 If the neighbourhoods cover the whole net-works and the associated graph is connected, then the dy-namics that applies (28) when the neighbourhood Nj is selected ensures global convergence towards σSMC: - Deterministically, when all the neighborhoods are se-lected cyclically;
- In probability and in expectation, when all the neigh-borhoods are selected randomly from a fixed probability distribution {qj= P[Nj] > 0|Pjqj= 1}, independently at each time.
Proof. The density operators with support on HSMCare the unique common fixed states for all the ENj maps: if
there is at least a neighborhood Njin which the reduced state has support outside HNj,SMC, so it cannot be
in-variant for the corresponding ENj. Consider
Vm(ρ) = 1 − Tr(ΠSMC,mρ),
where ΠSMC,mis the orthogonal projection on HSMCfor m qubits. It shall be proved by induction on the number of subsystems m that there exists a subsequence of T maps:
Eo
m= ENj(T )◦ . . . ◦ ENj(1),
such that for any ρ with support outside of HSMC: ∆Vm(ρ) := V (Emo(ρ)) − V (ρ) < 0. For m = 2 and Eo
2 = E{1,2}, the result is immediate given the structure (27): if a state has support outside of HrmSM C, it must have support on span{|01i, |10i}. As-sume that q = h01|ρ|01i + h10|ρ|10i > 0. Then the effect of the randomizing feedback map is that of transporting the state inside HSMC, so that:
∆Vm(ρ) := Vm(E2o(ρ)) − Vm(ρ) = −q < 0. Now assume that the inductive hypothesis holds for m subsystems, with some sequence of neighborhood asso-ciated to a map Eo
m. If another subsystem is added, the effect of Eo
mon this m + 1 multipartite system is evalu-ated. By the inductive hypothesis:
∆Vm(ρ) := Vm(Emo(ρ)) − Vm(ρ) < q < 0. This means that the trace of the output state onto HSMC,m has increased. The subspace HSMC,m⊗ Hm+1 has thus also gain probability. This subspace is spanned by |00 . . . 0i ⊗ |xi, |11 . . . 1i ⊗ |yi, with x, y ∈ {0, 1}. If the map E{m,m+1} is applied, anything in this subspace is mapped onto HSMC,m+1. Thus it also holds that
∆Vm+1(ρ) := Vm+1(Em+1o (ρ)) − Vm+1(ρ) < q < 0. In this way it has been proved that a sequence of maps that contracts Vmfor any m does exists. By considering cyclic application of the sequence associated to the map Eo
m+1, a straightforward application of the invariance principle (in the discrete-time version [18]), asymptotic convergence is achieved.
Convergence in probability is then proved via Lemma 1 in the Appendix, using the sequence associated to Emo, and convergence in expectation directly from conver-gence in probability. 2 The probability distribution over the neighborhoods can actually be made time dependent, as long as all of its el-ements stay bounded away from zero (see e.g. the proofs of convergence of [20]).
4 A Toy Example
Consider three systems of dimension two, labelled as 1, 2, 3, and allow for controlled interactions on neigh-borhoods {1, 2} and {2, 3}. The three algorithms for symmetrization that are compared, identified by the acronyms below, are respectively using:
(GOS): The gossip-type interactions (9); (DSC): The Dicke-preparing interactions (22); (SMC): The SMC-preparing interactions (28).
For all the above, alternating actions on neighborhoods N1= {1, 2}, N2= {2, 3}, are considered. The presented simulations illustrate the different behavior of the dy-namics proposed in the preceding sections. An initial state is generated by randomly selecting its spectrum and then randomly choosing a unitary orthonormal ba-sis in which it is diagonal, and the evolution of the purity, along with some relevant state populations, are plotted as in Figures 1-3.
Fig. 1. (Color online) Purity and population dynamics for the Dicke states induced by the gossip-type algorithm GOS.
The main observations are summarized in the following.
• Given the theoretical analysis of the previous sections, the final state for the SSC algorithm has support only on the span of the four symmetric Dicke vectors, and thus the evolution of the four respective populations is depicted in the figure. Similarly for the SMC algorithm and the |000i, |111i vectors.
• Figure 2 shows how the average SSC-attaining algo-rithm associated to maps of the form (22) saturates the population of the Dicke states, overall improving the purity of the state. It is worth noting that the increase is not monotone. The natural comparison is with the gossip-type algorithm (9), which also attains
Fig. 2. (Color online) Purity and population dynamics for the Dicke states induced by the SSC algorithm
Fig. 3. (Color online) Purity and population dynamics for the |000i, |111i states induced by the SMC algorithms.
average SSC: as illustrated in Figure 1, by projecting the state on the group commutant [19,20] with unital operations, the purity decreases while the populations of the Dicke states are invariant. While the above be-havior is typical for random initial states, if the initial states is e.g. pure its purity will not typically be pre-served even for the new SSC algorithm, unless it is a state that belongs to one of the eigensubspaces of S. • The SMC algorithm symmetrizes the state by
“push-ing” it in the subspace generated by |000i, |111i. The increase of the population of these subspaces is de-picted in Figure 3, along with the associated improve-ment in purity. A direct comparison with the other
algorithms shows that, for the given initial state, this method leads to the highest final purity, with a final value of 0.51 for SMC, versus 0.36 for the SSC and 0.17 for GOS. This is to be expected: in fact, the SSC dynamics leave all the eigenspace of the operator to be averaged invariant, while the SMC leaves invariant only the ones corresponding to maximum and minimal eigenvalue. Preparing only these two while preserving the global expectation allows not only for a better pu-rity, but for a perfect correlation of local measurement outcomes [19].
5 Conclusions
Two classes of QL dynamics that attain asymptotic av-erage consensus for networks of quantum systems have been introduced. The first improves the existing gossip-type dynamics, as in general it attains a purer out-put state while still guaranteeing SSC consensus. This is achieved by selecting pure representatives for each eigenspace of the global observable whose average has to be preserved. While the new algorithm cannot be simply obtained by randomized permutations of subsystems, as the gossip ones do, it is suitable for a simple discrete-time feedback implementation, in the spirit of [4]. The second dynamics are the first proposal of dynamics at-taining the stronger σSMC consensus, thus not only im-prove the purity of the output with respect to the gossip-type dynamics, but actually attain a gossip-type of consensus that is more similar in spirit to classical consensus: if the local observable σ is observed for one subsystem and gives a certain outcome, it is guaranteed that all other subsystems will indeed return the same outcome in sub-sequent measurements. This second method is also suit-able for feedback implementation. Both dynamics en-sure convergence in probability when i.i.d. randomiza-tion of the applicarandomiza-tion of maps is used, being thus ro-bust with respect to the ordering and suitable for unsu-pervised implementation. The i.i.d. requirements can be easily relaxed, following e.g. [20]. Remarkably, the pro-posed SSC-attaining dynamics can be used, in combi-nation with few additional resources, to asymptotically prepare and stabilize Dicke states in multipartite quan-tum systems. This can be seen as a discrete-time version of the conditional stabilization protocols proposed in [26] and the results of [13] for continuous time quantum semigroups. Similarly, a strategy can be devised to uti-lize the SMC-attaining dynamics to prepare GHZ-type states with proper initialization of the network. As a last remark, the presented convergence analysis is one of the first stability results for time-inhomogeneous sequences of CPTP maps using Lyapunov techniques, and could be of inspiration for more general convergence analysis for this important class of dynamics.
Acknowledgements
F.T. gratefully acknowledges Lorenza Viola and Peter D. Johnson for joint work on related topics, which made the relevance of Dicke states for consensus apparent, and L. Mazzarella, A. Sarlette and M. Mirrahimi for insightful comments on this work. Work partially supported by the QFuture research grants of the University of Padova, and by the Department of Information Engineering research project QUINTET and QCOS.
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A Discrete-time Invariance Principle
The main tool we are going to use in proving convergence of the SSC algorithm is LaSalle’s invariance principle, which we recall here in its discrete-time form [18].
Theorem 4 (Discrete-time invariance principle) Consider a discrete-time system x(t + 1) = T [x(t)]. Suppose V is a C1
function of x ∈ Rn, bounded below and satisfying
∆V (x) = V (T [x]) − V (x) ≤ 0, ∀x (A.1)
i.e. V (x) is non-increasing along forward trajectories of the plant dynamics. Then any bounded trajectory con-verges to the largest invariant subset W contained in the locus E = {x|∆V (x) = 0}.
B Randomization Lemma
The following Lemma allows us to conclude the conver-gence in probability of the proposed methods, as soon as they admit a finite sequence that ensures a strict con-traction of the a Lyapunov function.
Lemma 1 (Convergence in probability) Consider a finite number of CPTP maps {Ej}Mj=1, and a (Lya-punov) function V (ρ), such that V (ρ) ≥ 0 and V (ρ) = 0 if and only if ρ ∈ S, with S some density operator set. Assume furthermore:
(1) For each j and state ρ, V (Ej(ρ)) ≤ V (ρ); (2) There exists a finite sequence of maps
Eo= EjK◦ . . . ◦ Ej1, (B.1)
with j`∈ {1, . . . , m} for all `, such that V (Eo(ρ)) ≤ V (ρ) for all ρ 6= S.
Assume that the maps are selected at random at each time t, with independent probability distribution Pt[Ej] at each time, such that exist some ε > 0 for which Pt[Ej] > ε for all t. Then, for any γ > 0, the probability of having V (ρ(t)) < γ converges to 1 as t converges to +∞.
Proof. LaSalle-Krasowsii invariance theorem (in discrete time, [18]) ensures us that by repeating the map Eo in-finite time would entail convergence to ρd, and hence V (ρ) → 0. Thus, for any fixed γ, application of the se-quence (B.1) for a large enough number of times Nγ > 0 would ensure V (ENγ
o (ρ)) ≤ γ for any ρ.
The proof is concluded by noting that the probability that a randomly selected sequence of B · Nγ · K maps does not contain Nγ contracting subsequence (B.1) is at most (1 − εNγ·K)B. The latter converges to 0 as B, and
